Question 83
There are pebbles at the center of each cell on a $12 \times 12$ board and two players, Painter and Flicker, play the following game.
Painter chooses a number of colors and paints each pebble with one of the colors. Flicker chooses two pebbles of the same color that has no blocking pebble in between them, and replaces one of the pebbles with the other and takes the replaced pebble. Flicker does this until there are no eligible pair left. Painter takes the remaining pebbles. Whoever has the bigger number of pebbles wins. What is the minimum number of colors needed for Painter to win regardless of how Flicker plays?
Explanations: we assume that the pebbles are sufficiently small and that they are placed at the exact center of the cells. That is, for pebbles denoted $x, y$ and $z$, we say that $y$ is blocking $x$ and $z$ if the center of the $y$'s cell is on the segment joining the centers of $x$'s cell and $z$'s cell. If we call Flickers action flicking, then in the following example, red pebbles can be flicked, green ones cannot be flicked and some yellow and some blue pebbles can be flicked and some cannot.
Painter chooses a number of colors and paints each pebble with one of the colors. Flicker chooses two pebbles of the same color that has no blocking pebble in between them, and replaces one of the pebbles with the other and takes the replaced pebble. Flicker does this until there are no eligible pair left. Painter takes the remaining pebbles. Whoever has the bigger number of pebbles wins. What is the minimum number of colors needed for Painter to win regardless of how Flicker plays?
Explanations: we assume that the pebbles are sufficiently small and that they are placed at the exact center of the cells. That is, for pebbles denoted $x, y$ and $z$, we say that $y$ is blocking $x$ and $z$ if the center of the $y$'s cell is on the segment joining the centers of $x$'s cell and $z$'s cell. If we call Flickers action flicking, then in the following example, red pebbles can be flicked, green ones cannot be flicked and some yellow and some blue pebbles can be flicked and some cannot.